3 Answers
3

Your series can be re-written in terms of the q-polygamma function $\psi_q(z)$ which is simply the logarithmic derivative of the q-gamma function $\Gamma_q(z)$. Both of which are special functions related to the theory of q-series: $$\sum_{n=0}^\infty\frac{1}{2^n+1}=\frac{\psi_{1/4}(1)-\psi_{1/2}(1)-\ln(3)}{\ln(2)}-\frac{3}{2}$$

Also as a consequence of several papers written by Erdos your sum is irrational. Erdos investigated similar series' when studying and also proving the irrationality of an analogous convergent series known as the "Erdős–Borwein constant" - the sum of the reciprocals of all the Mersenne numbers.

Now in terms of a different "closed form" then the one at the top that uses the q-polygamma function. I would say that it's unlikely you're going to find another such representation in terms of anything other then a similar type of q-series based special function. Though similar lambert series' and q series expressions can take some very nice closed form values when their input is evaluated at exponentials of scaled values of $\pi$, like the series given in the link provided by Mhenni Benghorbal.

Also on a somewhat unrelated note, the inner partial sum appearing in the third series representation I gave for your sum: $$\sum_{d\mid n}(-1)^{d}=\frac{1}{2}(-1)^n\sum_{a^2-b^2=n}_{(a,b)\in \mathbb{Z}^2}1$$
Is equivalent to one half of negative one to the $n^{th}$ power multiplied by the number of representations of $n$ as a difference of the squares of two integers.